Numerical investigation into characteristics of far-field aerodynamic noise radiated by inter-coach space of high-speed maglev trains

Abstract

Self-sustain oscillation leads to aerodynamic noise at inter-coach space. It can be seen from existing literature that the aerodynamic noise at inter-coach space has already become an important branch in the domain of vibration and noise reduction of high-speed wheel-rail trains. However, to the best of our knowledge, approaches focusing on the identical problem that does exist in the windshield area of high-speed maglev train have not yet been published. In this paper, a computational model is established for the inter-coach space of high-speed maglev train running at the speed of 600 km·h⁻¹. The model includes two half-mid-train with one outer full windshield connected. On the basis of Realizable k-ε model and Spalart-Allmaras DDES model, respectively, the steady and unsteady flow field are initially solved, and the pressure fluctuation on rigid surfaces, the vortex shedding process and the distribution of sound source power are then visualized to make clear the genesis mechanism of aerodynamic noise in the windshield area. With penetrable integration surfaces constructed, the far-field sound radiation is obtained from the Ffowc Williams-Hawkings (FW-H) equation, and the contribution rate of the quadrupole noise sources to the total aerodynamic noise is then calculated as a comparison to the previous conclusions that aerodynamic noise at inter-coach space of high-speed trains was mainly dipole. Of great significance, it is concluded in this paper that at the speed of 600 km·h⁻¹, the quadrupole sound source contributes about 35.6% to the total sound energy in the far field, which is lower than the percentage using the entire train model at the same speed level but still cannot be ignored. The results in this paper are able to provide guidance for future researches on vibration and noise reduction of high-speed maglev train.

Keywords

High-speed maglev train aerodynamic noise inter-coach space penetrable integration surface quadrupole effect

Introduction

In 1922, The German Hermann Kemper proposed the concept of electromaglev. With the advance of the industrial revolution and the further speed increase of the traditional wheel-rail trains encounter bottlenecks, Germany, Japan,¹ the United States, South Korea,² China³ and many other countries have carried out research on high-speed maglev and medium and low speed maglev technology. The existing lines include Shanghai Maglev Line, Yamanashi test line, Incheon Airport Maglev, etc. The high-speed maglev train overcomes the limitations of the wheel-rail adhesion and the bow network relationship, and becomes the fastest ground vehicle, effectively making up for the speed gap between high-speed wheel-rail train and aircraft. Because of its characteristics of low noise, low pollution and flexible route selection, it is ideal for the “point-to-point” medium and long-distance transportation trunk between the urban centers of big cities, which greatly saves the intercity commuting time of passengers. Although high-speed maglev railway has obvious advantages compared with other means of transportation, it also has some disadvantages that have been criticized, of which sound pollution produced at higher running speeds may to some extent affect the health and well-being of residents near the line.⁴ The running noise of high-speed maglev train is mainly composed of propulsion noise, structural noise due to guideway vibrations and aerodynamic noise resulting from airflow moving past the train. The aerodynamic noise is proportional to the sixth to eighth power of the running speed, which will become the dominant noise source as the running speed increases further.⁵ On the other hand, the aerodynamic noise propagates to the interior of the coach through doors, inter-coach space, etc., resulting in serious internal noise problems, which will significantly affect the ride comfort of passengers. Therefore, it is necessary to study the genesis mechanism of aerodynamic noise at inter-coach space of high-speed maglev train, so as to provide guidance for future researches on vibration and noise reduction.

Attempts have already been made to investigate the characteristics of aerodynamic noise of high-speed trains. The main test methods include field test, full-scale model wind tunnel test and scale model wind tunnel test. The aerodynamic noise of the flow field around the train can be divided into the noise directly generated by the fluid when interacting with the rigid surface and the noise generated by the vibration caused by the fluid on the structure, which is going to be further studied.⁶ One of the main difficulties of aeroacoustics is that the sound source and propagation are located in the same physical medium, making it uneasy to distinguish the source area of pressure fluctuation from the propagation area of sound pressure fluctuation. Much effort has been put into determining the main aerodynamic noise source of the high-speed train. Gautier et al.⁶ succeeded in identifying the source by field test using the trackside acoustic antenna which consists of a microphone array, and the output signal of the microphone was processed to locate the source, demonstrating the fact that the head car was the main aerodynamic noise source of a TGV. In a similar manner, Mellet et al.⁷ concluded that the aerodynamic noise generated by the head bogie of a TGV, which was mainly broadband, contributed more than that by the other bogies to the total aerodynamic noise. Yamazaki et al.⁸ found that the inter-coach space was the main low-frequency noise source of high-speed trains based on a wind tunnel test of the 1:5 Shinkansen train scale model. Nagakura⁹ also pointed out that the bogies, the pantograph head, the door, the wiper, et al. were main aerodynamic noise sources of Shinkansen trains by means of scale wind tunnel test. In a word, earlier studies made clear that the head car bogie, the pantograph, the inter-coach space, the bogie skirt boards, et al. were main aerodynamic noise sources of a high-speed train. On the basis of these studies, some numerical approaches and line tests on vibration and noise reduction techniques of high-speed trains were further carried out. Noh et al.^10,11 studied the applications of noise mitigation measures experimentally on Korean high-speed trains (KTX) running at 300 km/h and reduced the sound pressure level (SPL) of aerodynamic noise by 8–10 dB and 3∼8 dB with the installation of fairing and bellows, respectively. Zhang et al.¹² made discussion on the orientation, installation positions and various structures of pantograph fairing, studied the influence of bogie skirt boards as well, and finally designed a low-noise structure for a full-scale high-speed train, abating the average SPL by 3.2 dB compared to that of the original train.

Such researches made advances in locating the aerodynamic noise source of high-speed trains, and since it was concluded that the inter-coach space had the characteristics of low-frequency noise, many further researches on the genesis mechanism of low-frequency noise and the corresponding active and passive noise control measures at inter-coach space, specifically, were carried out. Frémion et al.¹³ arranged sound sensors in the windshield area and carried out a field test on a TGV train running at the speed of 350 km/h, finding that the low-frequency noise was particularly obvious and radiated around. He et al.¹⁴ used the exterior sound source identification system to analyze the sound source distribution law and spectrum characteristics of high-speed train. For the measure point A 7.5 m away from the track center line, the weighted SPL of noise generated at inter-coach space of high-speed train reached 107.0 dBA and 109.0 dBA when the train ran at the speed of 341 km/h and 386 km/h, respectively. Cui et al.¹⁵ investigated a two-mid-train model with inner windshields connected through detached eddy simulation (DES) and found that the recess and back of the inter-coach space was the main noise source. Liu et al.¹⁶ numerically verified that the full windshield generated less aerodynamic noise than the half windshield at the running speed of 350 km/h through large eddy simulation (LES) and Ffowc Williams-Hawkings (FW-H) acoustic analogy methods.¹⁷ Dai et al.¹⁸ designed two types of outer windshield and confirmed through simulation that the full-windshield form was able to lessen the overall SPL on the sides of near-field by about 13 dB. Zhao et al.¹⁹ carried out a test in a 1:15 model wind tunnel, discovering that the turbulent fluctuating pressure at inter-coach space could be significantly reduced and the vibration and noise reduction could be achieved by means of full windshield treatment or setting spoilers (spoiler balls, spoiler pillars) upstream of the inter-coach space. Similarly, Mizushima et al.^20,21 also verified the effectiveness of two passive noise control methods based on wind tunnel tests: setting spoiler thin line upstream and rounding the downstream edge. Wang²² obtained the noise spectrum of cavity noise at inter-coach space at different running speeds under the condition that the outer windshield only contained the upper opening or the lower opening through line tests, while the control measures of peak frequency and peak amplitude were numerically simulated and further discussed by adding extra components.

Compared with traditional wheel-rail trains, researches on aerodynamic noise of maglev trains commenced later, and current literature mainly focused on the total aerodynamic noise based on geometric simplification rather than concentrating on specific parts and components. In terms of field test, the earlier research can be traced back to the noise level measurement of TR08 train by Barsikow et al.²³ on the test line in Germany. Zhao et al.,²⁴ Duan et al.²⁵ conducted field measurements of the noise of high-speed maglev train and medium and low-speed maglev train respectively in order to supply foundation for environmental assessment. In terms of numerical simulation, Zhou et al.²⁶ captured the vortex structure characteristics of the train shoulder, high curvature part and wake area through improved delayed detached eddy simulation (IDDES). Based on the research of Tan et al.²⁷ on the key problems of quadrupole noise of high-speed trains, Zhang et al.²⁸ studied the aerodynamic excitation characteristics of 600 km·h⁻¹ high-speed maglev train, and found that the quadrupole sound source was mainly concentrated in the streamline and wake area of the tail car, which had low-frequency characteristics. Foreseeingly, Yu²⁹ took high-temperature superconducting (HTS) maglev train in low-vacuum tube as a research object, of which the running speed could reach as high as 1000 km·h⁻¹, and studied the influences of vacuum degree, running speed, blockage ratio et al. on the distribution of aerodynamic noise sources. However, Zhou, Zhang and Yu all adopted the simplified geometric model of maglev train, which failed to highlight the characteristics of geometric irregularity, notably, at inter-coach space.

It can be seen from existing researches that the aerodynamic noise at inter-coach space has already become an important branch in the field of high-speed wheel-rail train. However, to the best of our knowledge, approaches to solve the identical problem that also exists in the field of high-speed maglev train has not yet been published. In order to meet higher design speeds, existing high-speed maglev trains have widely adopted outer full windshield at inter-coach space, which is ideal for noise suppression. However, neither smooth curve transition¹⁶ nor the form that contains openings²² is able to remove the cavity structure. Also, the assumption that aerodynamic noise at inter-coach space of high-speed trains was mainly dipole no longer applies to the case where the running speed increases further. High-speed wheel-rail trains and high-speed maglev trains differ greatly in their configurations and running speed range. Consequently, the aerodynamic noise characteristics at inter-coach space of high-speed maglev train is also worthy of being studied.

In this paper, the aerodynamic noise characteristics at inter-coach space of high-speed maglev train running at the speed of 600 km·h⁻¹ is studied numerically. Firstly, the numerical calculation model is established with a detailed definition of the geometric model, computational domain, turbulence model and boundary conditions, which is mainly reflected in The aerodynamic noise calculation model section. Secondly, by solving the flow field, the cloud diagram of time gradient root mean square of fluctuating pressure is demonstrated, vortex structures based on Q-criterion are displayed, and the distribution of acoustic power level is determined. The genesis mechanism of aerodynamic noise at inter-coach space of high-speed maglev train is then analyzed. This part is mainly reflected in the Flow field and sound sources at inter-coach space section. Thirdly, FW-H acoustic analogy methods are used to obtain the solution of far-field aerodynamic noise. Rigid integration surface and a combination of rigid integration surface and penetrable integration surface constructed near the train body are adopted respectively and the results are compared so as to estimate the contribution rate of quadrupole sources to the sound energy in the far field. This part is mainly reflected in the Characteristics of far-field radiated noises section.

Theoretical review of cavity noise genesis mechanism

The acoustic radiation at high Reynolds numbers generated by turbulent flow is initially attributed to the broadband excitation of the cavity. In other words, this means that the turbulent boundary layer excites the cavity at different frequencies, because the size of the turbulent structure varies from micro scale to macro scale.^30,31 Then broadband noise is generated. However, the exposed flow may be laminar and generate oscillations from the cavity inlet due to the instability of the mixing layer between the flow above the cavity and the initially stationary air in the cavity. When the vibration of the swirling structures resonates with the cavity, the noise is intense,³² and the peak value appears on the pressure fluctuations spectrum. Resonance occurs when the Strouhal number St which represents the vibration frequency of vortex structure hangs the cavity resonant frequency

f_{H}

, while the former is function of the flow Mach number M, according to the following Rossiter formula

S t = \frac{f L}{U_{e}} = \frac{n - γ}{M + \frac{U_{e}}{U_{c}}}

(1)

where

U_{e}

is the outflow velocity of the boundary layer,

U_{c}

is the convection velocity of the vortex structure, n is the oscillation mode, values of γ averaged over the speed range are given in Table 1. For discussions in this paper, γ = 0.25.

Table 1.

Values of γ to length/depth ratio in Rossiter formula.³²

L/D	4	6	8	10
γ	0.25	0.38	0.54	0.58

The vibration of these structures produces a periodic fluid volume passing through the inlet surface of the cavity. Like a piston, a Helmholtz resonator is formed with the volume of the cavity. Among the various definitions of Helmholtz frequency, the following one is chosen³³

f_{H} = \frac{c_{0}}{2 π} \sqrt{\frac{S}{d^{'} V}}

(2)

where

d^{'} = d + 2 d_{e}

is the effective length of the fluid in the neck,

d_{e}

refers to end corrections on either side, S is the opening section, V is the cavity volume,

c_{0}

is sound velocity. Some authors have proved the piston effect through numerical calculation,³³ especially using the FW-H equation.¹⁶ This is the most general form of Lighthill³⁴ equation, essentially, an inhomogeneous wave equation, which takes into account the boundary effect of moving solid and predicts the aerodynamic noise of moving object in fluid.

The solution of this equation³⁵ is obtained by the convolution product of Green’s function. For the typical case of subsonic cavity flow, the formal solution of FW-H equation is

\begin{array}{l} 4 π p^{'} (x, t) = \\ \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} \iint_{f > 0} \frac{δ (g)}{r} T_{i j} d V d τ - \frac{δ}{δ x_{i}} \iint_{f = 0} \frac{δ (g)}{r} L_{i} d S d τ \end{array}

(3)

The part on the left represents the propagation of sound. For the right side, each item can be considered as sound source items. The first one contains the Lighthill stress $T_{i j}$ , $T_{i j} = ρ u_{i} u_{j} - e_{i j} + δ_{i j} (p - c_{0} ρ)$ . $e_{i j}$ is viscous stress. It behaves like a quadrupole and represents the volume source. The second item comes from the instantaneous resistance with dipole behavior on the cavity wall. For impenetrable surfaces like the train surface, $L_{i} = p n_{i}$ . The unipolar item produced by the piston effect in the neck of the chamber is not typical in cavity flows, so it is neglected. The numerical results³⁶ show that the far-field radiation and the near-field radiation are strongly generated by the dipole method at low Mach numbers. In the far-field, the acoustic energy is concentrated on the fundamental frequency and the first harmonic of the coherent structure. Among different noise sources, the strongest noise seems to come mainly from the resonance between the cavity and Kelvin-Helmholtz oscillation.

The aerodynamic noise calculation model

Geometric model

In this study, the inter-coach space of a specific type of full-scale high-speed maglev train is selected as the research object, and the outer full windshield is adopted at inter-coach space between two half-mid-trains, as shown in Figure 1(a). The length of each half-mid-train is $l = l_{c} + l_{w} + l_{s} = 12.25 m$ , including the net length of the coach $l_{c} = 11.84 m$ and the half of the width of full windshield $l_{w} = 0.40 m$ . The outer full windshield adopts a smooth curve transition. In order to meet the needs of the train passing through the curve, there is a slit in the middle of the windshield, and the width of the slit is $2 l_{s} = 0.02 m$ , as shown in Figure 1(b). In order to facilitate daily maintenance and shock absorber installation,²² lower openings of specific specifications are set on the bogie skirt board below the outer full windshield, as shown in Figure 1(c). The maglev train runs on a T-shaped guideway. The distance between the top of the train and the guideway surface is $h = h_{c} + h_{s u s} = 3.18 m$ , including the net height of the coach $h_{c} = 3.17 m$ and the suspension gap $h_{s u s} = 0.01 m$ . The width $w_{c} = 3.43 m$ , while the width of the upper surface of the guideway $w_{g} = 2.10 m$ , as shown in Figure 1(d).

Figure 1.

Geometric model of maglev train. (a) isometric view, (b) windshield, (c) lower opening, (d) front view.

Computational domain and boundary conditions

The computational domain for the high-speed maglev train is depicted in Figure 2. The length of the computational model is L = 24.5 m, the height H = 40 m and the width W = 40 m. Maglev train usually run on the elevated line. Therefore, the train and the guideway are located in the middle of the computational domain, and the coaches extend to the boundaries in the front as well as in the back. The implication here is that the flow field in the computational domain is not affected by the streamlines at the head train and the tail train. The cross-section ABCD right in front of the first mid-train is the inlet boundary which is set as velocity inlet condition. Velocity is 166.6667 m·s⁻¹ (equivalent to a running speed of 600 km·h⁻¹) when the computation is conducted. The cross-section KLMN right behind the second mid-train is the outlet boundary which is set as pressure outlet condition with one standard atmospheric pressure. The right, left, top and bottom sides are defined as symmetric boundaries so that the normal velocity of which is 0. The influence of wall surface on the flow field is eliminated while ensuring the full development of the flow field. The surface of the maglev train is defined as fixed boundary with no-slip condition. To simulate ground effect, the guideway is defined as a slipping surface, with a speed equal to the running speed of the train. The high-speed sections of maglev lines are mostly located in the suburbs, so sound barrier is not considered in the model. Since the train runs in open air, there is no significant sound reflection, so the boundary conditions are appropriate for aeroacoustic simulation.

Figure 2.

Computational domain of maglev train, (a) stereo view, (b) left view.

Mesh generation

The ANSYS ICEM CFD 19.0 pre-processing software is used to conduct structural mesh generation on the surface and surroundings of the maglev train, and the mesh distribution is shown in Figure 3. The maximum size of the mid-train is 500 mm. In order to accurately describe the aerodynamic noise sources of the surface dipole, a boundary layer is added on the surface of the maglev train, as can be seen from Figure 3(b). The boundary layer has 10 layers. The thickness of the first layer is 5 mm and the stretching ratio is 1.2. The total number of grid cells is about 4.8 million. The minimum mesh quality is 0.482 using determinant 3 × 3 × 3, which meets the practical needs of engineering. The mean values of Y-plus on train body surface are all around or less than one, indicating that the grid size meets the basic requirements of DDES solution.

Figure 3.

Computational grids for inter-coach space model.

Mathematical and physical model

The ANSYS Fluent 19.0 software is used to for both steady and unsteady calculation. In the simulation calculation, the train speed is 600 km·h⁻¹, the Mach number is about 0.5, and the air compressibility effect is obvious. Therefore, for the calculation of steady flow field, the density-based implicit solution method is adopted, and Realizable k-ε turbulence model is selected for numerical simulation. The convection term and dissipation term are discretized by second-order upwind scheme. Then, DDES model based on the simple Spalart-Allmaras^37–39 model is adopted for unsteady calculation. For the calculation of unsteady flow field, the modified turbulent viscosity is discretized by second-order upwind scheme, and bounded second-order implicit scheme is used to discretize the time derivative. The main modeling schemes adopted for the CFD simulations are summarized in Table 2.

Table 2.

Main modeling schemes adopted for the CFD simulations.

Time dependency	Steady	Unsteady
Turbulence model	Realizable k-ε	DDES
Solver	Density based	Density based
Flux type	Roe-FDS	Roe-FDS
Gradient discretization	Least squares cell based	Least squares cell based
Flow discretization	Second-order upwind	Second-order upwind
Turbulent kinetic energy discretization	QUICK
Turbulent dissipation rate discretization	Second-order upwind
Modified turbulent viscosity discretization		Second-order upwind
Transient formulation		Second-order implicit

To ensure the characterization of the behavior with a maximum frequency of 20 kHz and a frequency resolution of 10 Hz, the time step of unsteady calculation is set to 2.5 × 10⁻⁵ s with a maximum of 30 iterations in each time step, and 8 × 10³ time steps are calculated in total.

Flow field and sound sources at inter-coach space

Characteristics of flow structures

The intensity of the dipole sound source is directly related to the fluctuating pressure. Therefore, it is necessary to accurately simulate the pulsating flow field around the train. The key to simulating the pulsating flow field is to simulate the development of boundary layer and the vortex structure at inter-coach space. Due to the effect of air viscosity, a boundary layer with a large velocity gradient will be formed when air flows through the surface of the train body, and its thickness is defined as the vertical distance of the wall normal direction when the velocity magnitude is equal to $0.99 V_{i n}$ , where $V_{i n}$ represents the freestream velocity at the velocity inlet. The boundary layer cloud diagram of the horizontal section of the train at a speed of 600 km·h⁻¹ (166.6667 m·s⁻¹) is colored with the value of velocity magnitude, as shown in Figure 4.

Figure 4.

Boundary layer cloud diagram of horizontal section (600 km·h⁻¹).

It can be seen that when the maglev train runs at high speed, inter-coach space is a major disturbance source, which is caused by flow instability occurs within the space. More specifically, a flow-acoustic coupling mechanism characterized by recirculation and impingement of the shear-layer on the front surface of the downstream coach is formed, thus inducing significant aerodynamic noise.

In order to show the characteristics of transient flow field around the train, the second invariant of velocity gradient tensor, namely Q-criterion,⁴⁰ was used to identify the vortex structure around the train. The instantaneous three-dimensional vortex structure (Q = 15,000) around the train at the speed level of 600 km·h⁻¹ at three arbitrarily chosen moment (Time Step = 2400, 4150, 6000) is shown in Figure 5. As can be seen from Figure 5, outer full windshield restrains the vortex shedding and flow separation to some extent compared to the case where inner windshield was applied,¹⁵ but vortexes still appear in the vicinity of the windshield and diminishes quickly. The vortex shedding at the lower opening is relatively more intense and stretches forward along the flow direction, forming horseshoe-shaped vortexes. The generation of quadrupole sound source is usually accompanied by strong vortex shedding.⁴¹ Consequently, the identification of vortex structures is especially meaningful as it lays foundation for the positioning of the potential quadrupole sound sources.

Figure 5.

Isosurface of the instantaneous Q-criterion, (a) Time Step = 2400, (b) Time Step = 4150, (c) Time Step = 6000.

Surface pressure fluctuation

The main sources of aerodynamic noise in flow field are monopole, dipole and quadrupole. In the calculation of aerodynamic noise of high-speed maglev train, it is usually considered that the surface of the train body is rigid, and there is no significant displacement on the surface of the train in practice.²⁸ Therefore, the contribution of monopole sound source to aerodynamic noise was not considered. This paper mainly analyzes the dipole and quadrupole sound sources.

Let the surface of the train body be D , $D \in R^{3}$ . For any flow field, at any time t and any position x , the dipole sound source strength on the train surface can be represented by the time gradient root mean square of fluctuating pressure $p_{r m s}^{'}$ , and the expression is

p_{r m s}^{'} (x, t) = \sqrt{\frac{1}{T} \int_{t - \frac{T}{2}}^{t + \frac{T}{2}} {(p^{'} (x, ξ))}^{2} d ξ}, x \in D

(4)

where T is the sampling time. T is set equal to the time step of the unsteady calculation to ensure that

p_{r m s}^{'}

has converged for all frequency domain components of pressure fluctuation are considered. Assume that the flow field has already reached a steady state at

t = t_{0} - \frac{T}{2}

p_{r m s}^{'}

is an absolute value of time gradient root mean square of fluctuating pressure. In order to display the distribution of pressure fluctuation more intuitively, dimensionless processing was conducted on

p_{r m s}^{'} (x, t_{0})

to obtain its dimensionless value

C_{p^{'}}

, which can be expressed as

C_{p^{'}} (x, t_{0}) = \frac{p_{r m s}^{'} (x, t_{0})}{\max {p_{r m s}^{'} (w, t_{0}) | w \in D}}, x \in D

(5)

where w is an arbitrary position vector, the end of which may locate on any point on the train surface D .

The $C_{p^{'}}$ cloud diagram at the speed level of 600 km·h⁻¹ is shown in Figure 6. As shown in Figure 6(a), at the speed of 600 km·h⁻¹, almost all the pressure fluctuations with dimensionless value $C_{p^{'}}$ greater than 0.1 occur in the windshield area. Thus, sound pressure is generated in this part and radiates to the far field. It can be observed clearly in Figure 6(b) that the dipole sound source is distributed in a band in the middle and upper areas on the downstream side of the windshield, while on the upstream side the pulsation is very weak. Similarly, the pressure fluctuation at the lower opening is also mainly distributed at the downstream edge. Therefore, it can be concluded that when the high-speed maglev train runs at the speed of 600 km·h⁻¹, the downstream side of the windshield and the downstream edge of the lower opening are main dipole noise sources.

Figure 6.

Cloud diagram of dimensionless time gradient root, (a) aerial view of entire model, (b) zoomed-in view of windshield area and lower opening.

Distribution of sound source power

In spite of the importance of sound pressure as an important physical parameter of noise evaluation, the magnitude of sound pressure is directly related to the distance from the sound source and the environment in which the sound is measured. Sound source power is therefore introduced in order to measure the sound radiation capacity of a sound source. In many engineering applications concerns, turbulence generates radiated aerodynamic noise, which is mainly broadband, exhibiting a frequency spectrum covering a wide range of frequencies. Broadband noise source method refers to the analysis method based on a combination of the computation of steady flow field, semi-empirical formula and Lighthill noise analysis method, and since the method solves the statistical turbulent flow, it has the advantages of shorter calculation period and higher efficiency. In order to further determine the energy contribution of sound source of dipole and quadrupole of each part of the train, the equivalent sound source power value of each part will be determined.

Dipole sound source is produced by the interaction between object surface and air flow in the flow field, whose sound source radiation power is directly proportional to the sixth power of flow velocity. The expression for the radiated sound power of dipole source is,²⁸

P_{A d} = \int_{S} \frac{A_{c} (y)}{12 ρ_{0} π c_{0}^{3}} \bar{{[\frac{δ p^{'} (y)}{δ t}]}^{2}} d S (y)

(6)

where y is the spatial coordinate vector of the sound source,

A_{c}

is the associated area, S is the area of sound source.

Quadrupole sound source is produced by viscous stress caused by fluid interaction, whose sound source radiation power is directly proportional to the eighth power of flow velocity. Proudman⁴² replaced the delay differential with the effective synchronous covariance to derive an expression for the quadrupole sound source power radiated per unit volume of isotropic turbulent flow. Lilley⁴³ reconsidered adopting a delay differential to deduce an expression for the radiated sound power. The expression for the radiated sound power per unit volume of quadrupole source is based on both these approaches,¹²

P_{A q} = a_{ε} ρ_{0} ε {(\frac{\sqrt{2 k}}{c_{0}})}^{5}

(7)

where

a_{ε}

is a model constant, which was determined to be 0.1 by Sarkar and Hussaini⁴⁴ through direct numerical simulation (DNS) of isotropic turbulent flow, while the TKE k and the turbulence dissipation rate ε can be extracted directly from nodes in the steady calculation applying to Realizable k-ε turbulence model, which corrected the possible negative positive stress in the standard equation.^45,46 Then, acoustic power level per unit surface area

L_{w}

, which is the relative measurement of sound power per unit surface area and reference sound power, is defined in a similar manner as SPL, as follows

L_{w} = 10 l g \sum P_{A} + L_{w 0}

(8)

where the summation notation Σ indicates that monopole, dipole and quadrupole are added algebraically,

L_{w 0} = 120 d B

is the reference acoustic power level. Generally, sound source power varies from the magnitude of 10⁻¹²W/m² (hearing threshold at 28 m) to 10⁻⁶W/m² (rocket engine), with the acoustic power level varies from 0 dB to 180 dB, respectively. The value of

L_{w}

at each point on the train surface is calculated by equation (8), while its distribution contours for the entire model, the windshield area and the interior of the cavity are shown in Figure 7(a)−(c), respectively.

Figure 7.

Sound power contours for the inter-coach space model. (a) the entire model, (b) the windshield, (c) the interior of the cavity.

Figure 7(a) shows that aerodynamic sound power is relatively evenly distributed on the surface of mid-train, and the value of $L_{w}$ fluctuates slightly around an average of 111.4 dB. Drastic fluctuations of $L_{w}$ values mainly take place in the windshield area. It is clearly demonstrated in Figure 7(b). That a wide range of $L_{w}$ values are distributed on the surface of the outer full windshield according to a certain law. In general, the downstream side of the outer full windshield has higher values of $L_{w}$ than the upstream side, which is caused by the impact of the detached vortex generated at the upstream edge on the downstream half of the windshield. However, an exception lies near the upper edge of the lower opening, where the peak value of $L_{w}$ upstream is significantly higher than the downstream side, which reaches 177.3 dB and even become the most concentrated sound power of the entire vehicle. Due to the shielding effect of the outer windshield, the values of $L_{w}$ inside the cavity are mostly below 90 dB, as can be seen from Figure 7(c).

Thus, the main aerodynamic noises sources at inter-coach space of a high-speed maglev train running at the speed of 600 km·h⁻¹ are located at the downstream side of the outer full windshield and the vicinity of the lower opening. The upstream side of the windshield except the area near the lower opening is not a main aerodynamic noise source.

Characteristics of far-field radiated noises

Arrangement for measure points of aerodynamic noises

The far-field noise characteristics of high-speed maglev train are closely related to the longitudinal location of the observation point, especially when the geometric irregularities at inter-coach space are taken into consideration. The aerodynamic noise generated at inter-coach space decays rapidly in the upstream and downstream directions, and the far-field aerodynamic noise is thus “re-distributed” through substituting the refined model for the coarse one. In order to study the noise characteristics at inter-coach space, a row of observation points with equal distance distribution are set at the standard measure points 1–25 7.5 m away from the central track line, as shown in Figure 8. The red line represents the parallel line 3.5 m above the central track line, which is in the same direction as the x-axis. Measure point 1 to measure point 25 are set 7.5 m away from the red line, where the coordinate of measure point 1 in the x direction is x = 0.25, and each measure point is equidistant with the distance of 1 m. That is, the coordinate of measure point 2 in the x direction x = 1.25, and so on. The measure points 8, 13 and 18 marked in red represent the upstream, middle and downstream receivers respectively, so the spectrum analysis of SPL and will be carried out and analyzed separately. The other measure points only read the total A-weighted SPL without displaying the specific spectrum diagram.

Figure 8.

Schematic diagram of measure points of aerodynamic noises.

Construction of penetrable integration surface

Theoretically, as long as the integration surface is set large enough to include all disturbance sources, the loss of sound source energy can be avoided, but this method will raise extremely high demand for computing resources and affect the computational efficiency. Therefore, in order to achieve a balance between accuracy and efficiency, compromise is reached in a way that the rigid integration surface is extrapolated only in the region where quadrupoles may occur.²⁸ According to the analysis of the excitation characteristics of inter-coach space, vortex shedding exists in both the upstream and downstream of the windshield, but the downstream vortexes extend rearward more wildly. Therefore, the original integration surface in the windshield area is extrapolated to form a penetrable integration surface that wraps around the inter-coach space. At the same time, in order to avoid pseudo-sounds induced by mass penetration,²⁷ the closure surface is set 6 m downstream of the windshield, as shown in Figure 9(a). This form of narrow front and wide rear has been applied in numerical simulation^27,28 and verified by wind tunnel tests. According to the characteristics of the horseshoe shaped divergence of the vortex structure, the penetrable integration surface is constructed in the form of narrow upstream and wide downstream. As depicted in Figure 9, the integral surface is horn-shaped. Now that part of the sound source surfaces is extrapolated, the solution of the far-field radiated noise is obtained in a way that considering the penetrable surfaces where the train body is wrapped but still taking the train surface as the sound source surface in the rest.

Figure 9.

Sketch of horn-shaped penetrable integration surface. (a) left view, (b) front view.

Distribution of far-field noise at inter-coach space

At the speed of 600 km·h⁻¹, the aerodynamic noise at inter-coach space is mainly composed of dipole noise and quadrupole noise. The far-field solution of dipole noise can be obtained directly through the integral of the fluctuating pressure on the surface of the rigid body. That is, when the integration surface is set on the surface of the train body, the quadrupole sources are naturally filtered as the wall surface cannot be penetrated. Unlike dipoles, quadrupole noise cannot be calculated directly. It has been already stated that through integrating the newly constructed sound source surface which is a combination of rigid surfaces and penetrable ones, the effect of quadrupoles will be reckoned in. Thus, the contribution of quadrupole noise can be obtained indirectly by subtracting the far-field solution of the rigid integration surface from that of the new integration surface. Firstly, the far-field noise solution of the rigid integration surface is analyzed to obtain the relationship between the total A-weighted SPL and the location of the receiver as well as the SPL spectra of the representative measure points. Then, by comparing with the solutions of the new sound source surface, the contribution rate of quadrupole noise at inter-coach space of high-speed maglev train running at 600 km·h⁻¹ is estimated.

Figures 10–12 show the characteristics of far-field radiation noise generated by two-mid-train model. Figure 10 shows the relationship between A-weighted SPL and the longitudinal coordinate of the measure point. It can be seen from the blue curve that among all 25 measure points, the SPLs from measure point 4 to measure point 15 are roughly the same, but decay rapidly to both sides. The maximum SPL from measure point 4 to measure point 15 is located at measure point 6, reaching 128.932 dBA, which is only 1.704 dBA higher than 127.228 dBA of measure point 15. Since nine of the 12 measure points are located upstream of the central measure point 13 but only two are located downstream of the central measure point, it can be inferred that such type of outer full windshield radiates dipole noise strongly to the upstream and middle stream in the far field under the action of free incoming flow. This is consistent with the fact that the surface pressure fluctuation in Figure 6 is mainly distributed on the downstream side of the windshield for the downstream side of windshield faces vast area upstream and middle stream. The attenuation on both sides of the high SPL band is consistent with the property of the dipole sources. It is worth noting that the dipole noise SPLs of measure point 1 and measure point 25 are 123.625 dBA and 108.833 dBA, respectively. In fact, these two measure points are somewhat close to the front inter-coach space and the rear inter-coach space respectively. According to the SPL superposition theorem, the sound pressure level directly superimposed between measure point 1 and measure point 25 is still approximately 123.625 dBA, which is 5.307 dBA lower than that of measure point 6. On the other hand, it can be seen from the yellow curve that at all the points, the far-field noises computed by using the new integration surface are always stronger than that by using the rigid integration surface. Different from the attenuation on both sides of dipole, the SPL considering dipole and quadrupole simultaneously rises gently with the coordinate x at the upstream and reaches a maximum value of 129.837 dBA at the measure point 11, then the SPL decreases monotonically rapidly downstream. Similar to the dipole situation, the actual SPL at the measure points near the pressure outlet should be close to the measure points most upstream of the next two-mid-train model, thus resulting a difference of 2.586 dBA between the local maximum and local minimum. Therefore, it can be concluded that highlighting the geometric irregularity of inter-coach space in the model has a significant impact on the far-field aerodynamic noise solution. Under the condition of train formation, the local maximum of far-field aerodynamic noise induced by the cavity structure at inter-coach space of the mid-train can be approximately 2.586 dBA higher than the local minimum. Compared with the conclusion in literature 28, it can be found that such increment would add new peaks to the SPL-x curve of the simplified model of entire train.

Figure 10.

Far-field A-weighted sound pressure levels at different observation points.

Figure 11.

SPL spectra of far-field noise radiated from dipole noise sources (dipole) and a superposition of dipole noise and quadrupole noise (total). (a) SPL spectra for measure point 8 (dipole), (b) SPL spectra for measure point 8 (total), (c) SPL spectra for measure point 13 (dipole), (d) SPL spectra for measure point 13 (total), (e) SPL spectra for measure point 18 (dipole), (f) SPL spectra for measure point 18 (total).

Figure 12.

The percentage of aerodynamic noise energy of dipole and quadrupole noise at each measure point.

From the perspective of frequency domain components, Figure 11 clearly shows the spectra comparison of dipole sound pressure under one-third octave band and the total sound pressure considering quadrupole. Firstly, Figure 11(a), (c) and (e) show the SPL spectra induced by dipoles at measure point 8, 13 and 18, respectively. Secondly, by comparing with them with the case considering quadrupoles, that is as shown in Figure 11(b), (d) and (f), the change in SPL of far-field radiated noise in frequency domain after introducing quadrupole noise sources is analyzed. For measure point 8, the SPL of dipole noise is between 68.1 dBA and 93.4 dBA, while the peak reaches 98.6 dBA after taking quadrupoles into account; for measure point 13, the SPL of dipole noise is between 71.8 dBA and 95.7 dBA, while the peak reaches 101.7 dBA after taking quadrupoles into account; for measure point 18, the SPL of dipole noise is between 69.2 dBA and 89.8 dBA, while the peak reaches 95.0 dBA after taking quadrupoles into account. It is worth noting that after considering the influence of quadrupole noise sources, the sound pressure level in individual bands may decrease, that is, the blue solid lines in Figure 11(b), (d) and (f) are sometimes above the yellow columns, which could be explicated by the filtering effect of the grid.²⁸

In general, considering the effect of quadrupole in the calculation of far-field noise, the SPL at each measure point can be increased by 0.179 dBA to 4.999 dBA. Therefore, when the speed of maglev train reaches 600 km·h⁻¹, even at such sound source as inter-coach space which was assumed to be dipole in the previous researches, the contribution of quadrupole sound source to the sound energy of far-field noise cannot be neglected.

In decibels, the superposition of sound pressure levels is not a linear addition, however, the sound energy can be added directly. In order to more intuitively show the contribution of quadrupole to the radiated aerodynamic noise of each measure point in the far field, equations (9) and (10) are used to transform the sound pressure level of each measure point into sound energy

I_{d} = 10^{\frac{S_{d}}{10}}

(9)

I_{d, q} = 10^{\frac{S_{d, q}}{10}}

(10)

where

S_{d}

S_{d, q}

are the sound pressure level measured at each observation point when only dipole is considered and both dipole and quadrupole effect are taken into consideration, respectively, while

I_{d}

I_{d, q}

are corresponding absolute value of sound energy. Equation (11) is used to calculate the proportion of quadrupole noise in the radiated sound energy at each point in the far field, as follows

W_{q} = \frac{I_{d, q} - I_{d}}{I_{d, q}} \times 100 %

(11)

Figure 12 shows the percentage of dipole and quadrupole noise in the radiated noise energy of each measure point. Compared with the total contribution rate of 71.7% of quadrupoles to the far-field sound energy in literature 28, the percentage of quadrupole noise source at any of the measure point is no more than this amount, indicating that the windshield area does has the property of dipole sound source in a relative sense compared with the entire train model. It can be read from Figure 12 that among all 25 measure points, the contribution percentage of quadrupole noise sources is in the range of (10%, 50%) at 14, while less than 10% at three and in the range of (50%, 70%) at 8, indicating that quadrupole effect is negligible at most of the longitudinal coordinates, but also does not surpass the dipole at the majority of receivers. Further observation of Figure 12 also shows that among the eight measure points dominated by quadrupoles, except that measure point 1 has relatively large error due to the great influence of boundary conditions of the velocity inlet, measure point 18 to measure point 24 are distributed in the middle and downstream direction of inter-coach space. This is consistent with the backward extension of the horseshoe vortex. From measure point 5 with the highest dipole percentage to measure point 21 with highest quadrupole percentage, the contribution rate of quadrupole noise generally shows an upward trend, and only decreases slightly from measure point 13 to measure point 15. After measure point 21, the quadrupole noise decreases rapidly to less than 50% at measure point 25, which is related to the attenuation of vortex strength. The average value of quadrupole noise source contribution in the far-field sound energy is analyzed by four schemes: (a) including all measure points, (b) excluding measure point 1, (c) excluding measure point 1 and 2, (d) excluding measure point 1, 2 and 3. The improvements in scheme (b)∼(d) is consequences of the interference of the velocity inlet. Under the four schemes, the percentages of quadrupole noise in the far field are calculated as: (a) 36.07%, (b) 35.22%, (c) 34.94%, (d) 35.56%. In order to eliminate the interference of velocity inlet as much as possible, it is considered that at the speed of 600 km·h⁻¹, the total contribution rate of the quadrupole noise generated at inter-coach space of a high-speed maglev train to the far-field sound energy is about 35.6%, which is significantly lower than the result of the entire train model in literature 28. This is because the model in this paper does not include the region of train wake, which is the main quadrupole noise source of high-speed maglev train. The result is meaningful because it reveals that when the train speed is further increased, the geometric irregular structures of the body surface such as the windshield area can also produce quadrupole noise that cannot be ignored. Considering the possibility of further speed increase of high-speed wheel-rail train, the quadrupole noise generated by body structures such as pantographs and bogies is also worthy of further study.

Conclusion

This paper selects the two-mid-train model of high-speed maglev train with a speed of 600 km·h⁻¹ as the research object. The steady flow field is solved by using the Realizable k-ε model, and then the transient flow field is solved by using the Spalart-Allmaras model. Using the FW-H acoustic analogy method, the train body surface and the penetrable surface partially enveloping the vehicle body are selected as the sound source integration surface respectively. The characteristics of far-field noise radiated by inter-coach space of the high-speed maglev train are analyzed, and the following conclusions are drawn:

1. With the outer full windshield installed, there is still a certain degree of spatial disturbance at inter-coach space as the geometric irregularity is not completely eliminated. The dipole sound source distributed on the windshield surface is mainly located at the downstream side of the windshield and the downstream edge of the lower opening. The sound power is mainly distributed at the downstream side of the windshield and near the lower opening, and is most intense at the upper edge of the lower opening.

2. Compared with the simplified model of maglev train, highlighting the geometric details at inter-coach space has a significant impact on the numerical simulation results. Under the condition of train formation, the local maximum of far-field aerodynamic noise induced by the cavity structure at inter-coach space of the mid-train can be approximately 2.586 dBA higher than the local minimum, which would add new peaks to the SPL-x curve of the simplified model of entire train.

3. At the speed of 600 km·h⁻¹, the total contribution rate of the quadrupole noise generated at inter-coach space of a high-speed maglev train to the far-field sound energy is about 35.6%, which is significantly lower than the result of the entire train model but still cannot be ignored. The assumption that aerodynamic noise radiated by inter-coach space is mainly dipole is no longer applicable.

4. The research results of this paper are able to provide a reference for the noise control of high-speed maglev train at higher speed level. The penetrable integration surface constructed is able to capture the quadrupole sound sources, which can also provide a basis for the analysis of aerodynamic noise generated by the body structures of other vehicles such as high-speed wheel-rail trains.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Rusi Wang

Jingyu Huang

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